Re: [firedrake] Solve a Variational problem in a part of the domain
Hi Anna, It *was* interesting! How do you intend to compute the integrals against the Heaviside function? The Heaviside function is non-polynomial so can't be computed exactly by quadrature. You are right that our standard setup can't cope with this situation currently. What we really need are function spaces that are defined to be constant in an entire column. We should try to sketch out with David and Lawrence what might be required in the infrastructure to support this. I can see it being useful for many other free surface type problems as well. all the best --Colin On 13 August 2015 at 15:36, Anna Kalogirou <a.kalogirou@leeds.ac.uk> wrote:
Hi Colin,
I attached a two-page document, where the system of equations I was talking about in the previous email is (1a)-(1e). I am puzzled with equation (1b) really. I can also eliminate (1b) & (1d) and solve for the remaining (1a), (1c) & (1f), but in this case I still don't know how to deal with (1f) since there is an integral into the integral, which contains the unknown function lambda^{n+1/2}.
Thanks, Anna.
On 13/08/15 13:47, Colin Cotter wrote:
Hi Anna, Sounds interesting. Please could you provide a bit more detail?
all the best --Colin
On 13 August 2015 at 12:09, Anna Kalogirou <a.kalogirou@leeds.ac.uk> wrote:
Dear all,
I have a system of equations to solve, which involves three space dependent functions phi, eta, lambda and two constants/scalars Z, W. These need to be solved simultaneously because all the equations involve at least 2 unknowns.
How do I solve that, considering that one of the scalar equations includes a spacial integral of one of the (still unknown) functions? Is it best to define the scalars as Constants?
This problem goes away when I write down the system in a standard FEM formulation, introducing the mass matrix etc. In this case, it is clear that I could solve for that function and then consecutively solve each of the remaining 4 equations. It is not that obvious how I can do that using Firedrake, that is why I thought I would have to solve simultaneously, but then I have the problem described above.
Regards,
Anna.
On 06/08/15 11:04, Lawrence Mitchell wrote:
-----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1
Hi Anna,
On 06/08/15 10:37, Anna Kalogirou wrote:
Dear all,
I have a rather simple question but I would like to get some feedback from someone in the Firedrake team.
I am now working on the problem which includes solving a variational problem in a part of the domain only (a water wave problem which includes a floating body).
I can think of a couple of possible solutions on how to solve this: 1. Define two domains and solve the problem separately in each domain. However, I will have to deal with nonzero boundary conditions on the common boundary.
2. I prefer solving the problem in the whole domain, since most of the equations/functions are valid everywhere. Then I can define a Heavyside step function which will be 0 in one part and 1 in the part of the domain I am interested in (under the floating body). I will essentially write down a variational problem valid everywhere, but will actually be zero in a part of the domain.
Is the 2nd step a good approach? The question essentially is how to split a mass matrix M_kl which is defined everywhere, and solve and integral form in a part of the domain only. I think step two is a fine approach. However, note the following issues at present. The way you would have to do this currently is as follows:
Define a DG0 field to hold your indicator function
indicator = Function(DG0)
# Set it to 1 in the appropriate part of the domain indicator.interpolate(...)
# Now use this extra field everywhere when defining your variational # problem.
However, all your integrals are still over the whole domain, you just pick up lots of zeros.
Steadily climbing our todo list (and at an increasing pace, I feel), is the ability to define proper sub domains in a mesh, and then be able to perform integrals over them. Until that time, I think approach 2 is somewhat easier to do than approach 1.
Cheers,
LAwrence -----BEGIN PGP SIGNATURE----- Version: GnuPG v1
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Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds
http://www1.maths.leeds.ac.uk/~matak/
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www.cambridge.org/9781107663916
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--
Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds
-- http://www.imperial.ac.uk/people/colin.cotter www.cambridge.org/9781107663916
Hi, I was planning to define the heavyside function as a DG0 function. Any ideas about solving equation (1b), which contains a time-update for both a function and a scalar? Thanks, Anna. On 13/08/15 18:35, Colin Cotter wrote:
Hi Anna, It *was* interesting!
How do you intend to compute the integrals against the Heaviside function? The Heaviside function is non-polynomial so can't be computed exactly by quadrature.
You are right that our standard setup can't cope with this situation currently. What we really need are function spaces that are defined to be constant in an entire column. We should try to sketch out with David and Lawrence what might be required in the infrastructure to support this. I can see it being useful for many other free surface type problems as well.
all the best --Colin
On 13 August 2015 at 15:36, Anna Kalogirou <a.kalogirou@leeds.ac.uk <mailto:a.kalogirou@leeds.ac.uk>> wrote:
Hi Colin,
I attached a two-page document, where the system of equations I was talking about in the previous email is (1a)-(1e). I am puzzled with equation (1b) really. I can also eliminate (1b) & (1d) and solve for the remaining (1a), (1c) & (1f), but in this case I still don't know how to deal with (1f) since there is an integral into the integral, which contains the unknown function lambda^{n+1/2}.
Thanks, Anna.
On 13/08/15 13:47, Colin Cotter wrote:
Hi Anna, Sounds interesting. Please could you provide a bit more detail?
all the best --Colin
On 13 August 2015 at 12:09, Anna Kalogirou <a.kalogirou@leeds.ac.uk <mailto:a.kalogirou@leeds.ac.uk>> wrote:
Dear all,
I have a system of equations to solve, which involves three space dependent functions phi, eta, lambda and two constants/scalars Z, W. These need to be solved simultaneously because all the equations involve at least 2 unknowns.
How do I solve that, considering that one of the scalar equations includes a spacial integral of one of the (still unknown) functions? Is it best to define the scalars as Constants?
This problem goes away when I write down the system in a standard FEM formulation, introducing the mass matrix etc. In this case, it is clear that I could solve for that function and then consecutively solve each of the remaining 4 equations. It is not that obvious how I can do that using Firedrake, that is why I thought I would have to solve simultaneously, but then I have the problem described above.
Regards,
Anna.
On 06/08/15 11:04, Lawrence Mitchell wrote: > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > Hi Anna, > > On 06/08/15 10:37, Anna Kalogirou wrote: >> Dear all, >> >> I have a rather simple question but I would like to get some >> feedback from someone in the Firedrake team. >> >> I am now working on the problem which includes solving a >> variational problem in a part of the domain only (a water wave >> problem which includes a floating body). >> >> I can think of a couple of possible solutions on how to solve this: >> 1. Define two domains and solve the problem separately in each >> domain. However, I will have to deal with nonzero boundary >> conditions on the common boundary. >> >> 2. I prefer solving the problem in the whole domain, since most of >> the equations/functions are valid everywhere. Then I can define a >> Heavyside step function which will be 0 in one part and 1 in the >> part of the domain I am interested in (under the floating body). I >> will essentially write down a variational problem valid everywhere, >> but will actually be zero in a part of the domain. >> >> Is the 2nd step a good approach? The question essentially is how >> to split a mass matrix M_kl which is defined everywhere, and solve >> and integral form in a part of the domain only. > I think step two is a fine approach. However, note the following > issues at present. The way you would have to do this currently is as > follows: > > Define a DG0 field to hold your indicator function > > indicator = Function(DG0) > > # Set it to 1 in the appropriate part of the domain > indicator.interpolate(...) > > # Now use this extra field everywhere when defining your variational > # problem. > > However, all your integrals are still over the whole domain, you just > pick up lots of zeros. > > Steadily climbing our todo list (and at an increasing pace, I feel), > is the ability to define proper sub domains in a mesh, and then be > able to perform integrals over them. Until that time, I think > approach 2 is somewhat easier to do than approach 1. > > Cheers, > > LAwrence > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1 > > iQEcBAEBAgAGBQJVwzElAAoJECOc1kQ8PEYvGswIAN+Mr6hD+U0NmsnZAGminaq+ > j/SqmI+o7GlbXRSgPqEfrpyWwFiiFVN6DnPmTboRooZr3h93ZhGDhECQ/O12XKA/ > 00yNuCiDn4YBaXJamWBM6u8ILdZYGMyVNTG1JOaZZIMBoiIkrTyYpXatGApcE4sD > 4T+DoAYSUEY7u6MV4eJmRc1lzZEd8Rw43YP+Viy0itCs2jKV+HsDpvpvu1DU/FSQ > zMeE/cRR8It6aImk1L3OON8Zo2ZB0HNirK6YuD3sMkIoqLEYANnNl8qaeU3LL227 > 3TL4ddkALXQLGaYGo5ETgtMKI6afNDIsfjwGUcVRhTTxHM3GKOMQgWV6miNtgv4= > =wE25 > -----END PGP SIGNATURE----- > > _______________________________________________ > firedrake mailing list > firedrake@imperial.ac.uk <mailto:firedrake@imperial.ac.uk> > https://mailman.ic.ac.uk/mailman/listinfo/firedrake
--
Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds
http://www1.maths.leeds.ac.uk/~matak/ <http://www1.maths.leeds.ac.uk/%7Ematak/>
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-- http://www.imperial.ac.uk/people/colin.cotter
www.cambridge.org/9781107663916 <http://www.cambridge.org/9781107663916>
_______________________________________________ firedrake mailing list firedrake@imperial.ac.uk <mailto:firedrake@imperial.ac.uk> https://mailman.ic.ac.uk/mailman/listinfo/firedrake
--
Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds
http://www1.maths.leeds.ac.uk/~matak/ <http://www1.maths.leeds.ac.uk/%7Ematak/>
-- http://www.imperial.ac.uk/people/colin.cotter
www.cambridge.org/9781107663916 <http://www.cambridge.org/9781107663916>
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On 14 Aug 2015, at 11:55, Anna Kalogirou <a.kalogirou@leeds.ac.uk> wrote:
Hi,
I was planning to define the heavyside function as a DG0 function.
Any ideas about solving equation (1b), which contains a time-update for both a function and a scalar?
I haven't been fully following along, but it looks like the formulation in (1f) introduces, effectively, a global coupling. Is that right? Assuming you didn't have any implementation constraints, how would you go about solving this? Lawrence
Hi all, I have to solve a problem with the bilinear form similar to the following: a = ( v*u + dt*inner(grad(u),grad(v)) )*dx + ( v*dt*assemble(u*dx) )*dx The last term is essentially the product of the integral of u and the integral of test function v. Is it even possible to solve this in Firedrake? Best, Anna. On 14/08/15 15:43, Lawrence Mitchell wrote:
On 14 Aug 2015, at 11:55, Anna Kalogirou <a.kalogirou@leeds.ac.uk> wrote:
Hi,
I was planning to define the heavyside function as a DG0 function.
Any ideas about solving equation (1b), which contains a time-update for both a function and a scalar? I haven't been fully following along, but it looks like the formulation in (1f) introduces, effectively, a global coupling. Is that right? Assuming you didn't have any implementation constraints, how would you go about solving this?
Lawrence
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-- Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds http://www1.maths.leeds.ac.uk/~matak/
participants (3)
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Anna Kalogirou
-
Colin Cotter
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Lawrence Mitchell